This dissertation contains work from the author's papers [35] and [36] with coauthor Eyvindur Palsson. The classic Mattila-Sjolin theorem shows that if a compact subset of $mathbb{R}^d$ has Hausdorff dimension at least $frac{(d+1)}{2}$ then its set of distances has nonempty interior. In this dissertation, we present a similar result, namely that if a compact subset $E$ of $mathbb{R}^d$, with $d geq 3$, has a large enough Hausdorff dimension then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. These types of results on point configurations with nonempty interior can be categorized as extensions and refinements of the statement in the well known Falconer distance problem which establishes a positive Lebesgue measure for the distance set instead of it having nonempty interior / Doctor of Philosophy / By establishing lower bounds on the Hausdorff dimension of the given compact set we can guarantee the existence of lots of triangles formed by triples of points of the given set. This type of result can be categorized as an extension and refinement of the statement in the well known Falconer distance problem which establishes that if a compact set is large enough then we can guarantee the existence of a significant amount of distances formed by pairs of points of the set
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/114979 |
| Date | 08 May 2023 |
| Creators | Romero Acosta, Juan Francisco |
| Contributors | Mathematics, Palsson, Eyvindur Ari, Yang, Yun, Sun, Wenbo, Elgart, Alexander |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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